Class 9 Notes - Areas of Parallelograms and Triangles

Download PDF Worksheets
Start a Practice Test

Areas of Parallelograms and Triangles

In this chapter, you will learn how to find the area of parallelograms and triangles, understand the properties that relate to their areas, and solve problems using these concepts.

1. Area of a Parallelogram

  • Definition: A parallelogram is a quadrilateral with both pairs of opposite sides parallel.
  • Area Formula: Area = base × height
  • Explanation: The height is the perpendicular distance from the base to the opposite side (not the side length).
  • Example: If the base of a parallelogram is 8 cm and the height is 5 cm, then
    Area = 8 × 5 = 40 cm²

2. Area of a Triangle

  • Definition: A triangle is a three-sided polygon.
  • Area Formula: Area = (1/2) × base × height
  • Explanation: The height is the perpendicular distance from the base to the opposite vertex.
  • Example: If the base of a triangle is 10 cm and the height is 6 cm, then
    Area = (1/2) × 10 × 6 = 30 cm²

3. Properties and Theorems

  • Parallelograms on the same base and between the same parallels have equal areas.
  • Triangles on the same base and between the same parallels have equal areas.
  • The area of a triangle is half the area of a parallelogram on the same base and between the same parallels.

4. Solved Examples

  1. Example 1: Find the area of a parallelogram with base 12 cm and height 7 cm.
    Solution: Area = 12 × 7 = 84 cm²
  2. Example 2: Find the area of a triangle with base 15 cm and height 8 cm.
    Solution: Area = (1/2) × 15 × 8 = 60 cm²
  3. Example 3: Two parallelograms have the same base of 10 cm and are between the same parallels. If the height of one is 6 cm and the other is 6 cm, what can you say about their areas?
    Solution: Both have area = 10 × 6 = 60 cm². Their areas are equal.

5. Practice Questions

  1. Find the area of a parallelogram with base 9 cm and height 4 cm.
  2. A triangle has a base of 14 cm and a height of 5 cm. What is its area?
  3. Two triangles have the same base and height. If the area of one is 24 cm², what is the area of the other?
  4. If the area of a parallelogram is 56 cm² and its base is 7 cm, find its height.

6. Summary

  • Area of parallelogram = base × height
  • Area of triangle = (1/2) × base × height
  • Parallelograms/triangles on the same base and between the same parallels have equal areas.